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Decidability and undecidability of theories with a predicate for the primes

  • P. T. Bateman (a1), C. G. Jockusch (a1) and A. R. Woods (a2)
Abstract
Abstract

It is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure 〈ω; +, P〉, where P is the set of primes, is undecidable and, in fact, that multiplication of natural numbers is first-order definable in this structure. In the other direction, it is shown, from the same hypothesis, that the monadic second-order theory of 〈ω S, P〉 is decidable, where S is the successor function. The latter result is proved using a general result of A. L. Semënov on decidability of monadic theories, and a proof of Semënov's result is presented.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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